Investigation of VaporâLiquid Nucleation for Water and Heavy Water

13938

J. Phys. Chem. C 2007, 111, 13938-13944

Investigation of Vapor-Liquid Nucleation for Water and Heavy Water by Density Functional Theory Dong Fu*,† and Xiao-Sen Li‡ School of EnVironmental Science and Engineering, North China Electric Power UniVersity, Baoding, 071003, People’s Republic of China and Guangzhou Institute of Energy ConVersion, Chinese Academy of Sciences, Guangzhou 510640, People’s Republic of China ReceiVed: May 22, 2007; In Final Form: July 11, 2007

The excess Helmholtz free-energy functional for water and heavy water is formulated in terms of a local density approximation for short-ranged interactions and a Weeks-Chandler-Anderson approximation for long-ranged attraction. The molecular parameters are regressed by three methods: (1) fitting to bulk properties, (2) fitting to surface tensions, and (3) fitting to both the bulk properties and the surface tensions. Within the framework of density functional theory (DFT), the nucleation properties, including density profiles, number of particles, critical supersaturations, and nucleation rates, are investigated for both water and heavy water. The critical supersaturations and nucleation rates predicted by DFT with molecular parameters regressed by method (3) are satisfactory compared with the experimental data.

1. Introduction Nucleation is the first step in many phase transformations and plays an important role in central areas of science and technology such as materials science, atmospheric sciences, and biology.1-4 In particular, vapor-to-liquid nucleation of dipolar and associating fluids is of fundamental and practical importance in atmospheric studies, and much effort has been devoted in recent years to understanding the properties of critical nuclei in such types of systems. Nucleation phenomena in fluid systems have been traditionally analyzed using classical nucleation theory (CNT),5-9 which assumes that density fluctuations in a nucleating system have identical properties to those of the bulk phase. Although this approach has been successful in the prediction of critical supersaturations in simple nonpolar fluids, it tends to predict nucleation rates that are too high at high temperatures and rates that are too low at the lower temperatures because of inappropriate treatment of curvature contributions to the work of formation of the critical nuclei. Some modified classical nucleation approaches10-14 have been successfully applied to the investigation of nucleation properties for associating fluids (e.g., the nucleation rates of water and heavy water can be satisfactorily evaluated compared with the experimental data).11-14 Density functional approach15 was applied to the study of homogeneous nucleation by Cahn and Hilliard16 nearly 50 years ago. Later, Oxtoby and co-workers extended this approach to the study of nucleation phenomena in a variety of systems.17-34 In general, density functional theory (DFT) allows for a more realistic treatment of curvature effects on the free energy of formation of critical clusters and predicts a vanishing barrier to nucleation at the thermodynamic spinodal. Moreover, DFT can be used to calculate bulk and interfacial properties of the systems of interest. * To whom correspondence should be addressed. E-mail: D.F.: [email protected]; x.-s. L.: [email protected]. † North China Electric Power University. ‡ Chinese Academy of Sciences.

To date, most of the DFT studies on vapor-to-liquid nucleation have focused on the properties of simple nonpolar fluids. Whereas the applications of DFT to the nucleation of nonpolar fluids are now well described, those of the associating fluids are much less understood. Talanquer and Oxtoby extended DFT to the study of nucleation of associating fluids with a single association site.31 Recently, Talanquer used DFT of statistical mechanics in a square gradient approximation to analyze the structure, size, and work of formation of critical nuclei in selfassociating fluids.34 However, the mentioned DFT work was limited to the case of model-like associating particles instead of real associating fluids. The main purpose of this work is to extend DFT to the study of nucleation properties for real associating fluids such as water and heavy water. To this end, the molecules are considered to be of spherical structure, and the formulations proposed by Chapman et al.35 and Huang and Radosz36 are used for the free energy due to association. A local density approximation (LDA) and a Weeks-Chandler-Anderson (WCA) approximation37 are used for short-ranged interactions and long-ranged attraction, respectively, to build an equation of state applicable for bulk phase, vapor-liquid interface, and critical nuclei. 2. Theory In the absence of an external potential, the grand potential (Ω[F(r)]) of an inhomogeneous system can be expressed as eq 1:

Ω[F(r)] ) A[F(r)] - µ

∫ F(r) dr

(1)

where F(r) is the number density of molecules, and µ is the chemical potential. In a LDA, the Helmholtz free-energy functional of the spherical fluid (A[F(r)]) can be written as eq 2:

A[F(r)] ) Aid[F(r)] + Ahs[F(r)] + Aatt[F(r)] + Aass[F(r)] (2)

10.1021/jp073971a CCC: $37.00 © 2007 American Chemical Society Published on Web 08/29/2007

Vapor-Liquid Nucleation for Water and Heavy Water

J. Phys. Chem. C, Vol. 111, No. 37, 2007 13939

where Aid[F(r)], Ahs[F(r)], Aatt[F(r)], and Aass[F(r)] correspond to the ideal gas, hard-sphere repulsion, attractive interaction, and association contributions to the free energy of the system, respectively. The ideal gas contribution to the Helmholtz free-energy functional is known exactly by eq 3:

Aid[F(r)]) NkT[ln[F(r)Λ3] - 1]

(3)

where k is the Boltzmann constant, T is the absolute temperature, and Λ is the de Broglie thermal wavelength. The Helmholtz free-energy contribution due to hard-sphere repulsions is formulated in the Carnahan-Straling approximation as eq 4:38

Ahs[F(r)]) NkT

4η[F(r)] - 3η[F(r)]2

(4)

{1 - η[F(r)]}2

where η[F(r)] ) (π/6)F(r)d3 is the packing factor, and d is an effective hard-sphere diameter. On the basis of the BarkerHenderson theory,39 the relationship between d and σ, the softsphere diameter, can be expressed as shown in eq 5:

d/σ )

∫ {1 - exp[u(r)/T]}dr

(5)

where u(r) is the intermolecular potential between molecules, and T is the temperature. For the Lennard-Jones potential in a WCA approximation, we obtain eq 6.37

{(

uWCA(|r - r′|) ) - 4

σ12 σ6 12 |r - r′| |r - r′|6

)

|r - r′| e x2σ 6

|r - r′| > x2σ 6

(6)

Verlet and Weis40 proposed eq 7 to replace eq 5.

d 0.3837 + 1.068/T* ) σ 0.4293 + 1/T*

(7)

where T* ) kT/ is the reduced temperature, and is the dispersion energy parameter. Compared with eq 5, eq 7 avoids the numerical integration; hence, it is more applicable for the calculation of bulk, interfacial, and nucleation properties. The Helmholtz free-energy contribution due to association between particles is expressed as eq 8:35,36

Aass[F(r)] ) NkT[M lnχA(r) - MχA(r)/2 + M/2]

(8)

where M is the number of non-interacting association sites on each molecule, and χA(r) is the fraction of molecules not bonded at site A, which is given by eq 9;

1

χA(r) ) 1+

∑B F(r)χB(r)∆(r)

(9)

where ∆(r) ) ghs[exp(a/kT) - 1](d 3κa) is related to the association strength, a is the association energy, and d 3κa is a measure of the volume available for bonding of any two sites on different molecules. The radial distribution function for the hard-sphere fluid (ghs) is given by eq 10.

ghs ) {1 - 0.5η[F(r)]}/{1 - η[F(r)]}3

(10)

Finally, the Helmholtz free-energy functional due to the attractive interaction is calculated in a mean-field approximation (eq 11).37

Aatt[F(r)] )

1 2

∫ ∫ dr′ dr uWCA(|r - r′|)F(r)F(r′)

(11)

3. Results and Discussion Under the framework of DFT, the nucleation rates can be predicted by using the molecular parameters as input. To obtain the molecular parameters, one may use a routinely applied method (i.e., regress the molecular parameters by fitting to the experimental data of vapor pressures and liquid densities). Besides this method, the molecular parameters can also be regressed by fitting to the surface tensions or to both the bulk properties and the surface tensions. 3.1 Nucleation Properties Predicted by Using the Molecular Parameters from Bulk Properties. In the bulk state, Aid[F(r)], Ahs[F(r)], Aatt[F(r)], and Aass[F(r)] become eqs 12-15.

Aid ) ln(FΛ3) - 1 NkT

(12)

Ahs 4η - 3η2 ) NkT (1 - η)2

(13)

∫ 4πr2uWCA(r) dr

(14)

Aass ) M[lnXA - XA/2 + 1/2] NkT

(15)

1 Aatt ) F 2

Equations 12-15 construct the equation of state for uniform associating fluids; hence, the phase equilibria, chemical potential (µ), and the pressure (pcoex) for bulk phases at a given temperature can be determined according to the requirement that pressure and chemical potential in both phases should be equal. To accurately model the phase equilibria of water and heavy water using the constructed equation of state, one needs four adjustable molecular parameters: (i) the diameter of each spherical molecule (σ), (ii) the dispersion energy parameter (/ k), (iii) the association volume (κa), and (iv) the association energy a/k. We have regressed these four parameters by fitting to the experimental data of liquid densities and vapor pressures far from the critical point.41-44 Table 1 presents the molecular parameters and the corresponding average relative deviations for pressure (p%) and liquid density (Fl%), in the cases of association sites per molecule M ) 1, 2, 3, and 4. From Table 1 we find that two association sites per molecule yields the best vapor-liquid equilibria for water and heavy water; hence, in this work, M ) 2 is adopted for the prediction of surface tension and nucleation rates. Figure 1 presents the phase diagrams and the saturation pressures for water and heavy water in the case of M ) 2 and the comparison with experimental data. Results for the phase equilibrium can be used to derive information about the structure and properties of the interface between coexisting phases. In particular, density profiles for the liquid-vapor interface can be obtained by minimizing the grand potential (Ω[F(r)]) in an open system with a fixed temperature (T) and chemical potential (µ). The condition δΩ[F(r)]/δF(r) ) 0 leads to an Euler-Lagrange integral equation that can be solved via a standard Picard method, and then the

13940 J. Phys. Chem. C, Vol. 111, No. 37, 2007

Fu and Li

TABLE 1: Regressed Parameters for Water and Heavy Watera

water

1 2 3 4 1

σ 10-10 m 2.942 2.926 2.980 3.050 2.971

heavy water

2 3 4

2.947 2.997 3.066

M

a

n

n

i)1

i)1

k-1 K 526.26 333.34 386.30 436.28 569.10

a k-1 K 3191.29 2889.99 2391.07 1391.11 3159.94

κa

p%

F l%

0.0018 0.026 0.011 0.014 0.0004

7.96 2.22 2.53 5.01 8.16

2.21 1.26 1.80 1.03 3.01

T range K 283-573 283-573 283-573 283-573 276-573

337.87 408.894 428.894

3007.05 2468.19 1468.19

0.018 0.006 0.012

3.42 4.11 5.33

1.41 1.32 1.39

276-573 276-573 276-573

l p% ) ∑ |1 - pcal/pexp|/n × 100 and Fl% ) ∑ |1 - Fcal /Flexp|/n × 100, n is the data points. The experimental data are taken from refs 41-44.

Figure 1. Phase diagrams (inset) and the saturation pressures for H2O and D2O. Experimental data; H2O (b),41,42 D2O (O).43,44 Lines are calculated from the equation of state; H2O (s) and D2O (----).

corresponding vapor-liquid surface tension can be calculated from the following relation (eq 16);

γ)

∫-∞ [a[F0(z)] - F0(z)µ + p ∞

coex

]dz

(16)

where F0(z) is the equilibrium density profile, a[F0(z)]stands for the local Helmholtz free-energy density, and pcoex is the coexistence pressure. The values of the surface tension can then be used to determine the initial guess of the radius of critical nuclei and the prefactor of the nucleation rate. Moreover, it can be used to estimate the work of formation of critical nuclei as predicted by the classical nucleation theories and to compare with the predictions of DFT. When the equilibrium bulk properties and vapor-liquid surface tensions are available, the initial guess of the density profile for the vapor-liquid nucleus can be expressed as follows (eq 17);

F(r) )

{

Fl 0 e r e r 0 Fsv r > r0

Figure 2. Nucleation rates for H2O. Symbols indicate experimental data.11 Lines indicate predicted results, DFT (s) and CNT (----), with ∆ΩCNT ) 16πγ3/[3FlkT ln S]2. The input surface tension (γ) and liquid density (Fl) are from prediction; CNT (- - -) with the experimental γ and Fl as input, as also presented in the work of Wolk and Strey.11

techniques for the convergence of the iteration process and the determination of equilibrium density profiles F0(r) have been well documented in the work of Oxtoby and Evans.19 Once F0(r) is obtained, the work of formation and number of particles are calculated from eqs 18 and 19.

∆Ω[F0(r)] ) Ω[F0(r)] - Ω[Fnv] ne )

∫0∞ [F0(r) - Fnv]dr

where r0 ) 2γ/(kTFl ln S) is the initial guess of the radius of critical nuclei, S ) p/pcoex is the supersaturation, and Fsv is the molecule density for the supersaturated gas surrounding the liquid droplets. Because the density varies only in the r direction, the Euler-Lagrange equation becomes δΩ[F(r)]/δF(r) ) 0. The

(19)

Using the predicted work of formation, the nucleation rate is then given by eq 20;

J ) J0 exp

[

]

-∆Ω[F0(r)] kT

(20)

where J0 ) x2γ/(πM)Fsv /Fl stands for the prefactor, M is the molecular weight, and γ is given by eq 16. Figure 2 shows the predicted nucleation rates of water and the comparison with experimental data. One may find from this figure that at a given temperature the nucleation rate of water increases monotonously with increasing supersaturation, and at the five temperatures, DFT captures such a dependence of nucleation rates as a function of supersaturation. From 220 to 2

(17)

(18)



260 K, the maximum scales (∆ ) JDFT/J exp) are, respectively, 15, 4, 0.056, 0.0077, and 0.0025. The best agreement occurs at T ) 230 K. Taking into account that the nucleation rate depends exponentially on the work of formation and that any small error in the calculation of ∆Ω/(kT) will lead to large deviation, the DFT calculations for nucleation rates are good. Also presented in Figure 2 are the nucleation rates predicted by CNT. When the input surface tension (γ) and liquid density (Fl) are from prediction, the agreement of the predicted nucleation rates with the experimental data are far from satisfactory. It is because the molecular parameters in Table 1 do not give satisfactory surface tensions as compared with experimental data (e.g., at 260 K, the predicted surface tensions (in 10-3N/m) for water is 91.8, yet the corresponding experimental data is 77.5).11 Such an overestimation of surface tension will be translated into huge effects on the work of formation, and finally, the predicted nucleation rates are several orders of magnitude lower than those presented in the work of Wolk and Strey,11 in which the input γ and Fl are from experiments. Compared with CNT, the DFT approach is not so dependent upon the equilibrium vapor-liquid surface tensions. In the DFT approach, the surface tensions play two roles: (1) determining a suitable initial guess of the radius of critical nuclei so that the iterative process can be convergent, and (ii) determine the prefactor of nucleation rate. Actually, the accuracy of the surface tensions is unimportant in the first role, and it only has slight effects on the prefactor of nucleation rate. We also calculate the nucleation rates with Katz-SaltsburgReiss (KSR) theory,10 which has proved to be more precise than CNT for the nucleation properties of one-site associating Lennard-Jones fluids31 and n-alcohols.45,46 In KSR theory, the work of formation is expressed as eq 21; 2 ∆ΩKSR ) 16πγ3/{3FlkT ln[p1/pcoex 1 ]}

Figure 3. The same as in Figure 2 except for D2O.

(21)

where p1 and pcoex stand for the partial pressure of monomers 1 in supersaturated gas and in equilibrium bulk gas, respectively. The KSR theory is theoretically better than CNT because the partial dimerization, terpolymerization, etc. in the associating gas are taken into account. However, in this work, the KSR theory yields almost the same results as CNT in the five investigated temperatures. We find that, at very low temperatures, the molar fraction of monomers in equilibrium bulk gas (xe1) is very close to that in supersaturated gas (xs1); hence, p1/ is very close to the supersaturation (S). For example, in pcoex 1 the case of T ) 220 K(the reduced temperature Tr ) T/Tc ) 0.311, Tc ) 707.34 K is the predicted critical temperature for water), xe1 ) 0.9998. When S changes from 16.20 to 23.20, xs1 changes from 0.9963 to 0.9947 and p/1pcoex changes from 1 16.14 to 23.08. Figure 3 presents the predicted nucleation rates of heavy water and the comparison with experimental data and those from CNT. The information shown in this figure is similar to that shown in Figure 2. From 220 to 260 K, the maximum scales from the DFT prediction are 120, 0.013, 0.038, 0.04, and 0.077, respectively. The best agreement occurs at T ) 250 K. Figure 4 shows the predicted and experimental critical supersaturations for water and heavy water, corresponding to an onset nucleation rate of J ) 107 cm-3 s-1. Both the DFT and the CNT capture the dependence of critical supersaturations as a monotone decreasing function of temperature. The CNT significantly over-estimates the critical supersaturations, but the DFT predictions agree well with the experimental data.

Figure 4. Predicted critical supersaturations (Sc) for H2O and D2O (inset), corresponding to an onset nucleation rate of J ) 107 cm-3 s-1. Symbols indicate experimental data.11 Lines indicate predicted results; DFT (s) and CNT (----).

Figure 5 shows the predicted number of particles and the density profile across the nuclei, corresponding to an onset nucleation rate of J ) 107 cm-3 s-1. To make the density profiles for water and heavy water sufficient distinguishable, we use the logarithmic coordinates to show the density. One finds from the density profiles that the radius of the critical nucleus for heavy water is slightly larger than that for water at the corresponding critical supersaturation (see Figure 4). However, because the regressed molecule diameter of heavy water is larger than that of water (see Table 1), the critical cluster of water contains a larger number of molecules, as shown in the main plot in this figure. In general, DFT is able to capture all these phenomena, although it over-estimates the number of particles either for water or for heavy water.


Figure 5. Predicted number of particles and the density profile across the nuclei (inset) for H2O and D2O, corresponding to an onset nucleation rate of J ) 107 cm-3 s-1. Symbols indicate experimental data;11 H2O (b) and D2O (O). Lines are predicted from DFT; H2O (s) and D2O (----).

3.2 Nucleation Properties Predicted by Using the Molecular Parameters from Surface Tensions or from Both Bulk Properties and Surface Tensions. In the above section, we have shown the nucleation properties predicted using the molecular parameters regressed by fitting to the liquid densities and vapor pressures. Other than fitting to the bulk properties, the molecular parameters can also be regressed by other two methods; one is to fit the surface tensions, and the other is to fit both the bulk properties and the surface tensions. In this section, we use these two methods to fit the molecular parameters and then predict the nucleation rates. Figure 6 shows the calculated surface tensions for water and heavy water and the comparison with experimental data.11,41,47 The molecular parameters of water (σ/10-10 m ) 3.22, k-1/K ) 333.88, a k-1/K ) 2690.20, and κa ) 0.019) are regressed by fitting to the surface tensions from 220 to 573 K with the n |1 - γcal/γexp|/n × 100 ) averaged deviation γ% ) ∑i)1 2.89%; the molecular parameters of heavy water (σ/10-10 m ) 3.25, k-1/K ) 337.90, a k-1/K ) 3007.85, and κa ) 0.012.) are regressed by fitting to the surface tensions from 220 to 320 K with the averaged deviation γ% ) 2.12%. Figures 7 and 8 show the nucleation rates of water and heavy water predicted by DFT and CNT, respectively, with the abovementioned molecular parameters as input. From 220 to 260 K, the maximum scales ∆ ) JDFT/J exp are 10, 0.1, 0.017, 0.0029, and 0.002 for water and 0.002, 0.001, 0.00079, 0.001, and 0.00271 for heavy water. The best agreement for water and heavy water occurs at T ) 230 K and T ) 220 K, respectively. Compared with the results shown in Figures 2 and 3, the molecular parameters obtained by fitting to the surface tensions yields unsatisfactory results either in DFT or in CNT. The most important reason is that the liquid densities in the bulk phase are significantly underestimated. For example, at 260 K the calculated liquid density is Fl ) 784.39 kg/m3, whereas the experimental data is Fl ) 997.03 kg/m3. As the predicted nucleation rates are not satisfactory, the critical supersaturations and number of particles are not calculated using this set of molecular parameters.

Fu and Li

Figure 6. Surface tensions for H2O and D2O. Symbols indicate experimental data; H2O (b)11,41,47 and D2O (O).11,47 Lines are from DFT; H2O (s) and D2O (----).

Figure 7. Nucleation rates for H2O. Symbols indicate experimental data;11 Lines indicate predicted results; DFT (s) and CNT (----). The input surface tension (γ) and liquid density (Fl) are from prediction. The molecular parameters are obtained by fitting to the surface tensions.

To fit both liquid densities and surface tensions well, we set the hard-sphere diameter as d ) a - bT, hence there are five adjustable parameters in the optimization. Although this expression greatly differs with that shown in eq 5, it keeps the tendency that d must decrease with the increase of temperature. The molecular parameters of water are σ/10-10 m ) 3.51 0.00201T, k-1/K ) 330.03, a k-1/K ) 2940.2, and κa ) 0.0214, the averaged deviations are γ% ) 4.1% and Fl% ) 2.2%, respectively. The molecular parameters of heavy water are σ/10-10 m ) 3.58 - 0.00204T, k-1/K ) 336.3, a k-1/K ) 2969.4, and κa ) 0.0168, the averaged deviations are γ% )



Figure 9. Liquid density (9: H2O and 0: D2O), vapor pressure (2: H2O and 4: D2O), and surface tension (b: H2O and O: D2O) for H2O and D2O. Symbols indicate experimental data calculated by the equations presented in the work of Wolk and Strey.11 Lines indicate predicted results; H2O (s) and D2O (----).

3.7% and Fl% ) 2.3%, respectively. The calculated bulk properties and surface tensions (from 220 to 260 K) are shown in Figure 9. Figures 10 and 11 show the nucleation rates predicted by DFT and CNT with the above two sets of molecular parameters. From 220 to 260 K, the maximum scales ∆ ) JDFT/J exp are 9.7, 5.1, 0.43, 0.057, and 0.095 for water, and 6.04, 0.133, 0.11, 0.121, and 0.125 for heavy water, respectively. The best agreement for water and heavy water occurs at T ) 240 K and T ) 230 K, respectively. We find that once the bulk properties and the surface tensions are satisfactorily fitted, the DFT approach gives qualitatively good nucleation rates as compared with experimental data. The temperature dependence is also


Figure 10. The same as in Figure 2 except the molecular parameters are obtained by fitting to both the liquid densities and the surface tensions.


improved as compared with Figures 2 and 3. However, the nucleation rates are still significantly underestimated by CNT or KSR with the calculated surface tensions and liquid densities as input because, for both water and heavy water, the surface tensions are overestimated whereas the liquid densities are underestimated (as shown in Figure 9). Figure 12 shows the predicted number of particles and the critical supersaturations (Sc), corresponding to an onset nucleation rate of J ) 107 cm-3 s-1. Compared with Figures 4 and 5, we find both the number of particles and the critical supersaturations (Sc) are slightly improved by using the molecular parameters regressed from both the bulk properties and the surface tensions.


Fu and Li Natural Science Foundation of China (Nos. 20576030 and 20606009), the Program for New Century Excellent Talents in University (No. 06-06-0206), the Research Fund of Key Lab for Nanomaterials, Ministry of Education (No. 2006-2), the National High Technology Research and Development Program of China (No. 2006AA05Z319), and the key program of North China Electric Power University. References and Notes

Figure 12. Predicted number of particles and the critical supersaturations (Sc) (inset) for H2O and D2O, corresponding to an onset nucleation rate of J ) 107 cm-3 s-1. Symbols indicate experimental data,11 H2O (b) and D2O (O). Lines are predicted from DFT. H2O (s) and D2O (----). The molecular parameters are obtained by fitting to both the liquid densities and the surface tensions.

4. Conclusion In summary, a nonuniform equation of state for water and heavy water is constructed by using LDA for short-ranged interactions and WCA approximation for long-ranged attraction. In bulk phase, vapor-liquid equilibria are calculated using the bulk equation of state. The molecular parameters are regressed by three methods. The surface tensions are calculated under the framework of DFT. With the predicted bulk and interfacial properties as input, the nucleation properties for water and heavy water are investigated. Our results show the following. (1) By using the molecular parameters from bulk properties, DFT quantitatively predicts the nucleation properties for water and heavy water. When the molecular parameters are obtained by fitting to both the bulk properties and the surface tensions, the predicted nucleation properties are improved and become satisfactory as compared with experimental data. (2) Although the experimental nucleation properties for water and heavy water are very close to each other, the predicted results for these two isotopic waters obviously differ because of the difference in the regressed molecular parameters. In general, the predicted number of particles and nucleation rates of water are higher than those of heavy water at a given supersaturation and temperature, yet the predicted critical supersaturations of water are lower than those of heavy water at a given onset nucleation rate. (3) Among the three approaches (CNT, KSR, and DFT) applicable for the calculation of nucleation properties of associating fluids, DFT is the most accurate one. In general, the classical nucleation approaches are easy to use but they lack the ability to calculate bulk properties and surface tensions; hence, they seriously relied on the experimental data. However, DFT approach is able to predict the nucleation properties with only the molecular parameters as input. Acknowledgment. The authors appreciate help from professor Talanquer, V., the financial support from the National

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